Can we add a scalar and a vector? $a+b+\mathbf x = \mathbf y $?

Question

I'm given the equation $$ a+b+\mathbf x = \mathbf y $$ With the vectors $\mathbf x=(x_1,x_2,x_3)$, $\mathbf y=(y_1,y_2,y_3)$ and the two scalars $a$, $b$.

Is the following correct? As a vector equation: \begin{align} a+b+\mathbf x &= \mathbf y \tag 1\\ a + b + (x_1,x_2,x_3)&=(y_1,y_2,y_3) \tag 2 \\ (a+b+x_1,a+b+x_2,a+b+x_3)&=(y_1,y_2,y_3) \tag 3 \end{align} And the vector equation as three separate scalar equations: \begin{align} a+b + x_1 = y_1 \tag 4\\ a+b + x_2 = y_2 \tag 5\\ a+b + x_3 = y_3 \tag 6\\ \end{align}

Also, if it is illegal, does $a+b+\mathbf x = \mathbf y$ have any meaning or is it just nonsense?

Answer 1

Use notation that will be easily understood $$ (a+b)\left<1,1,1\right>+\bf{x}=\bf{y} $$

Answer 2

Others have already stated that the answer is no, but I want to reinforce why this is in a formal way. The problem here is that you are viewing scalar and vector addition as the same thing, when in fact they are not. Although we use the same symbol, "$+$", for both operations, they are not the same thing because they operate on different sets. Suppose we have a vector space $V$ over a field $K$ - then scalar multiplication is an operation between two scalars that outputs another scalar. Formally speaking, $+:F\times F\to F ~ ; ~ +:(a,b)\mapsto a+b$ given some $a,b \in F.$ Vector addition takes in two vectors and outputs another vector. But we happen to use the same symbol for it because of its similarities to scalar addition. So, $+:V\times V \to V ~ ; ~ +:(\mathbf{u},\mathbf{v})\mapsto \mathbf{u}+\mathbf{v}$ given some $\mathbf{u},\mathbf{v}\in V.$ But trying to do $a+\mathbf{u}$ isn't defined, since we don't have any addition operators that can add elements of different sets like $?:F\times V \to V ~;~ ?:(a,\mathbf{u})\mapsto a+\mathbf{u}$. It's about as sensible as trying to add apples and oranges, so to speak.